Reachability Is Not Enough: Selection Failure and Repair in Conditional Flow Matching

Conditional flow matching (CFM) learns a velocity field by regression, but success is judged by the distribution it generates. If a model class contains a correct generator, must population CFM select one? It need not. We construct a finite-dimensional affine class for independent straight-line CFM with Gaussian endpoints in which every admissible positive-definite matrix weight measurable from the regression input excludes the correct generator. For identifiable affine classes, we characterize when reweighting can select a prescribed field under specified information access and derive the exact minimum global spectral contrast required. Near this obstruction, even the best attainable asymptotic variance for the weighted-regression ratio estimator studied here diverges, whereas an endpoint-derived moment equation using the same observations remains regular even at the obstruction. Controlled finite-update experiments show CFM moving correct synthetic and pretrained generators away from their targets. In a structured family, CFM attains lower regression risk than likelihood fitting yet much worse endpoint KL; in a richer version, endpoint fitting substantially reduces CFM's KL under a prespecified 10% empirical regression-risk budget.